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Properties

Label 1-837-837.778-r1-0-0
Degree $1$
Conductor $837$
Sign $0.451 + 0.892i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (−0.939 − 0.342i)5-s + (−0.719 + 0.694i)7-s + (−0.104 − 0.994i)8-s + (0.669 + 0.743i)10-s + (0.719 − 0.694i)11-s + (−0.848 − 0.529i)13-s + (0.961 − 0.275i)14-s + (−0.374 + 0.927i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.241 − 0.970i)20-s + (−0.961 + 0.275i)22-s + (−0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (−0.939 − 0.342i)5-s + (−0.719 + 0.694i)7-s + (−0.104 − 0.994i)8-s + (0.669 + 0.743i)10-s + (0.719 − 0.694i)11-s + (−0.848 − 0.529i)13-s + (0.961 − 0.275i)14-s + (−0.374 + 0.927i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.241 − 0.970i)20-s + (−0.961 + 0.275i)22-s + (−0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (778, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2070764645 + 0.1273074990i\)
\(L(\frac12)\) \(\approx\) \(0.2070764645 + 0.1273074990i\)
\(L(1)\) \(\approx\) \(0.4685061432 - 0.1242086001i\)
\(L(1)\) \(\approx\) \(0.4685061432 - 0.1242086001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.882 - 0.469i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.719 + 0.694i)T \)
11 \( 1 + (0.719 - 0.694i)T \)
13 \( 1 + (-0.848 - 0.529i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.961 + 0.275i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.374 - 0.927i)T \)
43 \( 1 + (-0.0348 - 0.999i)T \)
47 \( 1 + (-0.615 - 0.788i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.997 + 0.0697i)T \)
83 \( 1 + (-0.0348 - 0.999i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.93205379755856810180524716504, −20.68908633421130318950774514008, −19.692158634178741842709179532471, −19.50747995757194667363857520202, −18.7980213994549971072521135938, −17.63774420883656565750080846228, −16.97316845066857198418128829720, −16.30807972700535382290498660561, −15.498866831108937224902421651176, −14.643996458778622471587384367370, −14.14593553632313785798418624272, −12.599615907271639073696484908243, −11.93833305851368731443825667764, −10.953307009520697615380995001986, −10.08103904721684874491735592299, −9.5575928478019809146378566238, −8.32888823312725890841550156696, −7.67515985664467248903188854917, −6.722992288544922791141334830800, −6.382205441486563754386842330588, −4.73610323527485068097706427539, −3.92087365343938653025397983120, −2.6712675310366163027281663937, −1.401338857967359694496346826518, −0.123155708317315029247750280501, 0.61767596589765043287127276677, 2.00887314678570089042579581553, 3.19308832590639449183051145910, 3.73533837999749034445078029159, 5.13528815474054743025465196629, 6.362574478130875434273504740607, 7.239831863223681912907408110908, 8.231573246228519168978285353657, 8.77035221457360325405209345068, 9.695679104304786743106381783957, 10.490069082876943162410425607, 11.57794812092879207641377271215, 12.24351479414637821523312981912, 12.58050140550396075583818271061, 13.95895346183915554094248412740, 15.1615341700159331309797784838, 15.84227660624758083457859909294, 16.55448333573909766230627548876, 17.2021007594172294692993107625, 18.31919533942660405231079392042, 19.124417883588961645567111039084, 19.487465307393340527773805038915, 20.21143335549927764911861062478, 21.17910423938147852630452388465, 22.0182024446281265132685358239

Graph of the $Z$-function along the critical line